Calendar for a lunar-solar sabbath system

Certain religions observe a sabbath one day out of every seven, most notably Jews and Christians. The majority of Christians observe Sunday, while Jews and some Protestant denominations observe Saturday. Our modern calendar gives primacy to the seven-day week, so Sunday and Saturday always occur in the same column of the calendar. Sunday is either the first or last column:

Saturday is either the last or second to last column:

Some people have argued that the ancient Jews did not observe sabbath on a strictly seven-day cycle, but observed sabbath on new moons and every seventh day thereafter until the next new moon. Such a calendar is called a lunar-solar calendar, because it depends on the sun for the daily cycle and the moon for the monthly cycle. On our modern calendar, such a pattern of observance looks like this:

That looks very irregular, in part because our modern months don't align with the phases of the moon. To contrive a more moon-centric calendar, we can alternate between months of 29 and 30 days to approximate an average month length closer to the moon's 29.53-day cycle:

The vertical highlights now start at the beginning of each month, which makes the structure much more apparent. But to get a calendar truly suited for lunar-solar sabbath observance, we'll have to abandon the primacy of the seven-day cycle. Here's a format that puts each month in its own set of four rows, with an extra day on the first row for the new moon, and an extra day on the last row for months with 30 days:

This is much more regular, but very different from the calendar we're used to.

While Sunday and Saturday sabbath observance is derived from Exodus 20:9–10, which specifies observing every seventh day, motivation for lunar-solar sabbath observance may be based, at least in part, on Genesis 1:14–19, which dictates tracking time via the sun, moon, and stars. Observing sabbath every seven days relies only on the sun, never on the celestial motion of the moon or stars. Lunar-solar sabbath observance adds dependence on the moon by resetting every new moon. (I'm not aware that such a system depends on the stars, except to know when to add leap months, which does not affect the cycle of sabbath observance. If you know a way it does depend on the stars, or of some other sabbath system, email me.)

The main motivation for lunar-solar sabbath observance seems to be that a seven-day cycle can be interrupted and forgotten. Whether our modern Saturday or Sunday are an even number of seven-day weeks since the sabbaths of the Children of Israel in the wilderness, or even further back to the seventh day of creation week, no one knows. A lunar-solar sabbath system, however, resets with an astronomical event visible worldwide (weather permitting).